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User blog:Alemagno12/Making an OCF, attempt 2
I got inspired by this blog post to make another OCF, or at least something similar to it. I will call my previous OCF the JKL OCF and this one the lambda notation. *@x is the rest of the ordinal inside of the function the @x is inside of, excluding ends of parentheses... *...and $x is a copy of that @x (if $x is empty, the last $x in an expression is replaced to 0) *@1 = @ and $1 = $ *A function F() eventually overgrows another function G() if there's some value y < α -> F(α) such that F(x) ≥ G(x) for all x ≥ y *If X is a diagonalizer of F(), @X = sup($,$F($),$F($F($)),...) *If X is a diagonalizer of F(), Y is a diagonalizer of G(), and F() eventually overgrows G(), then X > Y *If X > Y, then @X > $Y, λX > λY, λx(X) > λx(Y) for all x, λX(x) > λY(x) for all x, and λx(Ψ(y,X)) > λx(Ψ(y,Y)) for all x and y. *If λx(X) is inside of function F(y) = Ψ(y,x), then λx(X) = X. *Ψ(x,0) = Ψ(x) *Ψ(x) = Ψε0(x) *Ψx(0) = x *Ψx(y+1) = sup(Ψx(y),Ψx(y)Ψx(y),Ψx(y)Ψx(y)Ψx(y),...) *For limit ordinal y, Ψx(y,z) = sup(Ψx(y1,z),Ψx(y2,z),Ψx(y3,z),...) *To avoid ellipses in the following rules, Ψ(Ψ(a,b),c) (b > c) is shortened to Ψ(a,c) (and Ψ(Ψ(a,b),1) means Ψ(a,1) if b = 1), but only in the rules. *λ0 is the diagonalizer of the function F(x) = Ψ(x,0) *Ψ(x+1,1) is the diagonalizer of the function F(y) = ΨΨ(x,1)(y) *Ψ(0,x+1) = λx *Ψ(Ψ(@λ0(x+λ0),y),1) is the diagonalizer of function F(z) = Ψ(Ψ(Ψ($λ0(x)+z,y),1)) *Ψ(Ψ(@λ0(@2λ0),x),1) is the diagonalizer of function F(y) = Ψ(Ψ(Ψ($λ0($2y),x),1)) *Ψ(Ψ(@λx+1(@2λ0),y),1) (0 < x ≤ y) is the diagonalizer of function F(z) = λx(Ψ(z,y)), in Ψ(Ψ($F(z),y),1) *Ψ(Ψ(@λy(@2λx),x),1) (y ≤ x) is the diagonalizer of function F(z) = λz(Ψ($2y,x)), in Ψ($F(z),1) *If X is the diagonalizer of function F(), Ψ(Ψ(@λz(@2X),y),1) (z ≤ y) is the diagonalizer of function G(w) = Ψ($2λz(F(w)),y), in Ψ(Ψ($λz($2G(w)),y),1) *If none of the rules apply, Ψ(x,y) can be changed to F(x) while calculating the FS of the ordinal (but only while doing that), where F(z) = Ψ(z,y) The current limit of this function is Ψ(Ψ(Ψ(Ψ(...,3),2),1)). I could extend to up to Ψ(Ψ(α -> Ψ(0,α),1)), but this notation is already strong enough. If there is a problem in the definition, please leave it in the comments. This function has a WAY faster growth rate than the strength I intended the JKL OCF to have. It's also a good notation to compare SAN with, since both notations use the same concept: in SAN it's 1-separators, and in here it's diagonalizers. I have done some analysis with this notation and SAN, and the results are: *The limit of pDAN is Ψ(Ψ(Ψλ0(0),1)). *The limit of sDAN is Ψ(Ψ(Ψ(1,1)λ0ω,1)). *The limit of DAN is Ψ(Ψ(Ψ(1,1)ω,1)). *The limit of NDAN is Ψ(Ψ(Ψ(1,1)λ0ω,1)). *The limit of WDEN is Ψ(Ψ(Ψ(1,1)Ψλ0(0),1)). *The limit of mWDEN is >> Ψ(Ψ(Ψ(1,1)Ψ(1,1)ω,1)) (I am not sure how mWDEN works starting from (1:)(1:)(1:)...{1:`2}, but if my guesses are correct, the limit of mWDEN is Ψ(Ψ(ΨΨ(1,1)(0),1)). You can find the analysis of this notation and standard ordinal notation here. I will make the blog post of the analysis of this notation and SAN after I finish the analysis of this notation and standard ordinal notation, but I can also make it now, if people want. Category:Blog posts